Toss apple slices with lemon juice and cinnamon in a small bowl. Heat 1 teaspoon oil and 1 teaspoon butter in a medium nonstick skillet over medium-high heat. Add the apples and cook, stirring occasionally, until tender, about 5 minutes. Keep warm.

Mix 1 teaspoon herbes de Provence, salt and pepper. Place chicken between sheets of plastic wrap and pound with a meat mallet or the bottom of a small saucepan to a 1/2-inch thickness. Sprinkle the chicken on both sides with the seasoning mixture.

Heat 1 teaspoon oil and 1 teaspoon butter in a large skillet over high heat. Add half the chicken and cook until no longer pink in the center, 2 to 3 minutes per side. Remove to a platter and keep warm. Add the remaining 1 teaspoon oil and 1 teaspoon butter to the pan; heat over high heat. Cook the remaining chicken in the same manner.

Add broth, lemon zest, the remaining 1/8 teaspoon herbes and any accumulated juices from the chicken to the pan. Cook, stirring to scrape up any browned bits, until slightly reduced, about 3 minutes. Spoon the sauce over the chicken and serve with the sautéed apples.